Luneberg lens and process for producing the same

ABSTRACT

A Luneberg lens having a single-layer structure or a multilayer structure containing a plurality of layers having different dielectric constants, wherein the respective structure is produced by mixing a polyolefin resin and/or a derivative thereof with an inorganic filler having a high dielectric constant, the volume ratio of the polyolefin resin and/or the derivative thereof to the filler being 99 to 50:1 to 50, adding a foaming agent to the resulting resin mixture and then performing preliminary expansion, and molding the resulting pre-expanded beads; and wherein at least a foamed dielectric layer having a dielectric constant of 1.5 or more is formed using the pre-expanded beads that have been subjected to classification and selection such that f(A) satisfies the expression 0.0005≦f(A)≦0.1, where f(A) is represented by the equation: f(A)=σa/Aave, σa is the deviation of a gas volume fraction Ar in the foamed dielectric layer, and Aave is the average of the gas volume fractions Ars at positions in the foamed dielectric layer.

TECHNICAL FIELD

The present invention relates to a Luneberg lens that is used fortransmitting and receiving electromagnetic waves and that hassatisfactory electrical characteristics and a method of producing thelens.

BACKGROUND ART

A known dielectric lens for microwaves has been disclosed in, forexample, Japanese Unexamined Patent Application Publication Nos.3-179805, 5-334934, 6-6126, 8-167811, 9-130137, and 2002-197923 andJapanese Examined Patent Application Publication No. 56-17767.

Among these known techniques, Japanese Unexamined Patent ApplicationPublication No. 3-179805 discloses a lens composed of non-foamdielectrics. Japanese Examined Patent Application Publication No.56-17767 discloses a lens composed of foam dielectrics. Furthermore,Japanese Examined Patent Application Publication No. 56-17767, andJapanese Unexamined Patent Application Publication Nos. 5-334934,6-6126, 8-167811, 9-130137, and 2002-197923 disclose filler-containingfoam dielectrics.

Electrical characteristics principally required of receiving ortransmitting antennas are (1) gain (or G/T=gain/(noise) temperature) and(2) sidelobes. A Luneberg lens antenna, which is particularly used as amulti-beam antenna or an antenna for mobile communication, is requiredto have the same focal length, the same gain (or G/T), and the samesidelobe characteristics, regardless of the direction of electromagneticwaves.

Of the above-described characteristics (1) and (2), the sidelobecharacteristics are significantly important since they are particularlysusceptible to influences from adjacent satellites and other nearantennas or tend to affect other antennas. For example, the sidelobelevels of a receiving antenna are required to be equal to or belowvalues recommended in (1) Electronic Industry Assoc. of Japan (EIAJ)CPR-5101A or (2) International Telecommunication Union-R (ITU-R)recommendation (for receiving broadcasting satellite system (BSS)).

The sidelobe is, so to speak, noise and has a power of 1/100 or less ofthat of the main beam. Thus, the sidelobe is susceptible to variousfactors of the antenna. In particular, in a lens antenna through whichelectromagnetic waves transmit, the sidelobe is significantly affectedby slight variations of the dielectric constant in the layersconstituting the lens.

In the Luneberg lens, it is further difficult to control an antennapattern such as sidelobes. The Luneberg lens is composed of dielectricshaving a dielectric constant (∈) of 1 to 2, and to achieve suchdielectric constant, it must contain a gas such as air. The gas iscontained by foaming, which must be controlled to be uniform at anyposition in the lens. However, because of the dispersion of a foamingagent, thermal uniformity in heating a thick lens, and resin meltviscosity, it was difficult to produce a Luneberg lens exhibitinguniformity in terms of the sidelobe.

In particular, in the case where the dielectrics were composed of acomposite dielectric consisting of three components as in the presentinvention: an olefin resin, an inorganic filler having a high dielectricconstant, and a gas, it was very difficult to produce the dielectriccomposite having uniform dielectric constant at any position by mixingthese components because these components vary greatly in theirdielectric constants: 2 to 3, 100 or more, and 1, respectively. Becauseof such difficulty in combination of the difficulty to control thefoaming, it was difficult to produce a satisfactory Luneberg lens thatcan exhibit satisfactory sidelobe characteristics for electromagneticwaves from any direction.

If the size of the Luneberg lens is increased, the gain is enhanced anda sharp beam is achieved, which results in easily satisfying thespecified value of the sidelobes. However, in view of the installationplace of the antenna and ease of installation, the Luneberg lens must bereduced in size. A general-purpose antenna is required to be compact andto satisfy the required electrical characteristics.

The known lenses described in the Patent Publications and the like areclassified into filler-free lenses and filler-containing lenses. Thedisadvantages of these lenses are described as follows.

[Filler-Free Lens]

A typical Luneberg lens is composed of a plurality of foamed dielectriclayers prepared by foaming polystyrene (PS). However, in this lens, PShas a dielectric constant of 2.5, and each layer has a dielectricconstant of 1 to 2; hence, the expansion ratio is low. Specifically, ata dielectric constant of 1.2 or more, the expansion ratio is 5 or less.At a dielectric constant of 1.4 or more, the expansion ratio is 3 orless. At a dielectric constant of 1.65, the expansion ratio is 2 orless. In this way, the expansion ratio is very low. The expansion ratioof a typical foamed product is generally 20 to 50 times. At an expansionratio of 5 or less, it is difficult to conduct forming. Therefore, it isdifficult to produce a uniformly foamed product at such low expansionratios as described above. In order to constitute a Luneberg lens bycombining dielectric layers foamed at such low expansion ratios, it isnecessary to control the expansion ratio of each layer with an accuracyof 0.1 times. Therefore, it was very difficult to achieve a dielectricconstant as designed.

In a bead forming method among various foam-forming methods,pre-expanded beads are prepared in advance. In the case of PS foamedmaterials having a low expansion ratio, the beads are only slightlyfoamed at such step. Thus, it is difficult to produce beads having auniform expansion ratio: various beads are produced in a broaddistribution of expansion ratio, ranging from a non-foamed bead to afoamed bead having an expansion ratio of 10 or more. Therefore, auniform lens could not be produced.

Furthermore, at an expansion ratio of 2 or less, it was extremelydifficult to form a shape: it was almost impossible to produce layershaving uniform electrical characteristics and having a relativedielectric constant of 1.7 or more. The layer having uniform electricalcharacteristics and having a dielectric constant of 1.7 or more isseldom produced.

In view of the circumstances, in some known filler-free lenses, themiddle layer was designed to have a greater dielectric constant in orderto increase the expansion ratio of layers having higher dielectricconstants, with dielectric constants (∈) of 1 to 1.7, and not 1 to 2.Such design adversely affects gain and sidelobes, which results infailure of producing a lens having satisfactory electricalcharacteristics.

With respect to a layer having a dielectric constant of 1.7 or more,there were cases in which the layer was made by bonding PS beads with anadhesive or by bonding a mixture of PS beads and glass fibers or beadswith an adhesive. In this method, since the adhesive having a dielectricconstant of 2 or more is present between the beads, the uniformity ofthe dielectric constants is significantly impaired. Furthermore, sinceadhesives generally have high tan δ (dielectric loss) and causestransmission loss, only lenses having low electrical characteristics canbe produced.

In the case of lenses produced by such a difficult method, the yield isnaturally low, which results in increase of cost.

Furthermore, the Luneberg lens composed of foamed PS is disadvantageousbecause the foaming ratio thereof is extremely low, resulting in a highmass (i.e., it becomes heavy).

[Filler-Containing Lens]

Among the above-mentioned problems with the filler-free lenses, withrespect to the weight reduction and the production of a layer having adielectric constant of 1.7 or more, a proposed method is to add afiller, such as titanium oxide (refer to Japanese Unexamined PatentApplication Publication No. 6-6126).

However, in this method, it was difficult to actually produce a Luneberglens that could be used satisfactorily with respect to sidelobes andvariations in gain, although theoretically the expansion ratio could beincreased so as to achieve the weight reduction and the production of alayer having a dielectric constant of 1.7 or more. The reason for thisis that with the three components: an olefin resin, an inorganic fillerhaving a high dielectric constant, and a gas, a dielectric compositehaving uniform electrical characteristics cannot be made because thesignificant differences of their relative densities: 0.9, 4 to 5 make itdifficult to homogeneously mix the components; and because nonuniformmixing, which results in nonuniform electrical characteristics, iscaused by the significant differences in the relative dielectricconstants of these components: i.e., the dielectric constants of theolefin resin, the inorganic filler, and the gas are 2 to 3, 100 or more,and 1, respectively. Consequently, a dielectric composite having theuniformity of the electrical characteristics cannot be provided.

As described above, controlling the expansion ratio with high accuracyis significantly difficult. In the filler-containing system, since thedielectric constants of components other than the gas are very high,variations in dielectric constant due to variations in foaming arelarge, whereby even small variations in a low expansion ratio causeincomparably greater variations in dielectric constant in the dielectriccomposite, as compared with the filler-free system.

Furthermore, in the filler-containing system, it is more difficult toperform uniform foaming as compared with the case in the filler-freesystem since foam breakages are easily to occur because the fillersegments are present in thin resin films generated during foaming.

In short, with known techniques, it was difficult to produce afiller-containing foamed product having uniform electricalcharacteristics, in particular, a filler-containing foamed producthaving a low expansion ratio, since in the filler-containing system, itwas difficult to obtain uniform dielectrics as compared with the case inthe filler-free system, and moreover, it was significantly difficult toobtain uniform dielectrics with a foamed product as compared with thecase of non-foamed product.

DISCLOSURE OF INVENTION

Accordingly, it is an object of the present invention to provide aLuneberg lens satisfying the requirements for both gain and sidelobecharacteristics, having lightness in weight and high uniformity, andachieving a reduction in cost due to mass production.

To solve the above-described problems, the present invention provides aLuneberg lens having a single-layer structure or a multilayer structurecontaining a plurality of layers having different dielectric constants,wherein the respective structure is produced by mixing a polyolefinresin and/or a derivative thereof with an inorganic filler having a highdielectric constant, the volume ratio of the polyolefin resin and/or thederivative thereof to the filler being 99 to 50:1 to 50, adding afoaming agent to the resulting resin mixture and then performingpreliminary expansion, and molding the resulting pre-expanded beads; andwherein at least a foamed dielectric layer having a dielectric constantof 1.5 or more is formed using the pre-expanded beads that have beensubjected to classification and selection such that f(A) satisfies theexpression 0.0005≦f(A)≦0.1, where f(A) is represented by the equation:f(A)=σa/Aave, σa is the deviation of a gas volume fraction Ar in thefoamed dielectric layer, and Aave is the average of the gas volumefractions Ars at positions in the foamed dielectric layer.

The inorganic filler having a high dielectric constant preferably ismade of titanium oxide, titanate, zirconate, or a mixture thereof. Thetitanate is preferably barium titanate, strontium titanate, calciumtitanate, or magnesium titanate. The zirconate is useful for finelyadjusting the dielectric constant of titanium oxide and for adjustingthe temperature dependence of titanium oxide by mixing with titaniumoxide.

The pre-expanded beads used for constituting the foamed dielectric layerhaving a dielectric constant of 1.5 or more can be classified andselected by gravity separation or size classification.

The Luneberg lens is produced by a method including the steps of mixinga polyolefin resin and/or a derivative thereof with an inorganic fillerhaving a high dielectric constant, the volume ratio of the polyolefinresin and/or the derivative thereof to the filler being 99 to 50:1 to50; adding a foaming agent to the resulting resin mixture and thenperforming pre-expansion; classifying the resulting pre-expanded beadsby gravity separation or size classification; and forming the classifiedpre-expanded beads into a shape. The present invention also providesthis production method. The forming is performed by a bead foam-formingmethod.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a cross-sectional view of a lens according to an embodiment ofthe present invention.

FIG. 2 is a cross-sectional view of a lens according to anotherembodiment of the present invention.

FIG. 3 shows a test method for evaluating performance.

FIG. 4 shows the density distribution of foamed beads containing afiller.

FIGS. 5-1 to 5-8 each show the density distribution of each groupclassified by gravity separation.

FIGS. 6-1 to 6-8 each show the density distribution of each groupclassified by weight and size classification.

FIG. 7 shows the design concept for designing dielectric constants ofthe Luneberg lens.

BEST MODE FOR CARRYING OUT THE INVENTION

Embodiments of the Luneberg lens of the present invention will bedescribed based on the drawings. In the drawings, the same referencenumeral represents the same element. Redundant description is notrepeated. The ratios of dimensions in the drawings are not always thesame as those of the actual objects described in the respectivedrawings.

In FIG. 1, a spherical, multilayer Luneberg lens 1 is designed so thatdielectric constants at any portion satisfy the following equation∈=2−(r/R)² (see FIG. 7, where r represents the radius of a hemisphericalcore 1 a, R represents the radius of a different diameter hemisphericalcore 1 b _(-n)). An electromagnetic wave antenna includes the Luneberglens 1, position-adjustable primary feeds 2, a holder 3 for holding theprimary feeds 2, the holder 3 being capable of adjusting the elevationangle, and cover 4 that transmits electromagnetic waves.

FIG. 2 shows a semispherical Luneberg lens 5 which is combined with areflector 6 for reflecting electromagnetic waves. This electromagneticwave lens is also combined with a primary feed and a holder for holdingthe feed (which are not shown in the figure) at a predetermined positionso as to form an antenna.

The Luneberg lenses 1 and 5 shown in FIGS. 1 and 2 are each produced asfollows: Pre-expanded beads are made from a resinous mixture prepared bymixing a polyolefin resin and/or a derivative thereof with an inorganicfiller having a high dielectric constant described above, the volumeratio of the polyolefin resin and/or the derivative thereof to thefiller being 99 to 50:1 to 50. The pre-expanded beads are formed intodielectric layers. The resulting dielectric layers (in FIG. 1, twohemispherical cores 1 a and two of each different diameter hemisphericalshell 1 b ₋₁ to 1 b _(-n); in FIG. 2, a semispherical core 5 a anddifferent diameter hemispherical shells 5 b ₋₁ to 5 b _(-n)) arecombined to form the Luneberg lenses. With respect to at least thedielectric layer having a dielectric constant of over 1.5, thepre-expanded beads selected by subjecting to classification are used asa material in order that f(A) represented by the equation f(A)=σa/Aavedescribed above satisfies the expression 0.0005≦f(A)≦0.1. If the contentof the filler is 50 percent by volume or more, foam breakage tends tooccur, thereby making it difficult to achieve foaming at a desiredexpansion ratio.

The foam dielectrics are produced by a foam bead forming method.

The method of bead forming includes producing resin beads containing afoaming agent, foaming the resulting resin beads at a predeterminedexpansion ratio to form pre-expanded beads, charging the pre-expandedbeads into a mold, and introducing steam into the mold to performfoaming by heating. By employing the steam heating, even when a thickproduct is molded, the steam is introduced between the beads touniformly heat the beads, thus achieving uniform forming.

A method of producing the Luneberg lens of the present invention will bedescribed below.

(I) Materials Used

(1) Resin

Any of the polyolefin resins, such as polyethylene (PE), polypropylene(PP), and polystyrene (PS), may be used. These resins have low tan δ andcan be applied to forming foamed beads.

(2) Filler

Any inorganic filler having a high dielectric constant may be used. Inparticular, titanium oxide (TiO₂), titanates, zirconates, or a mixturethereof are preferable because of their high dielectric constant. Amongthe titanates, barium titanate (BaTiO₃), strontium titanate (SrTiO₃),calcium titanate (CaTiO₃), and magnesium titanate (MgTiO₃) are suitable.

(3) Gas

Air may be used as a gas to be used. However, the gas is not limited toair.

(II) Production Process

(1) Foamed Bead Forming Step

1) Mixing Resin with Filler

The polyolefin resin and the inorganic filler having a high dielectricconstant are kneaded at a predetermined ratio. The resulting mixture ispelletized into pellets composed of the resin mixture having asubstantially uniform concentration (distribution density) of theinorganic filler having a high dielectric constant. To performclassification, the concentration of the inorganic filler having a highdielectric constant is required to be uniform. The concentration ispreferably within a range of plus/minus 0.5% and more preferablyplus/minus 0.1% with reference to the designed concentration. The mixingof the resin and the filler is performed with a mixing apparatus such asa biaxial or uniaxial extruder, a mixer, a kneader, or a Banbury mixer.

The size of each pellet is preferably ¼ or less and more preferably 1/10or less of the wavelength of the electromagnetic wave used.

2) Pre-Expansion (Gas Filling)

The resulting pellets are charged into a foaming tank. A gas is injectedinto a solvent and is enclosed in the pellets under high temperature andhigh pressure. The pellets and the gas are uniformly mixed in thesolvent to the extent possible in order that the amount of the gasenclosed is constant.

The steps of items 1) and 2) may be simultaneously performed. In thiscase, polyolefin monomers, polymerization promoters, and fillers areuniformly dispersed in a solvent, and the gas filling is performed whileconducting polymerization so that pre-expanded beads are formed.

(2) Classification and Selection Step

The resulting pre-expanded beads are classified by gravity separation orsize/weight classification to obtain beads having target density anddensity distribution. A specific procedure of the classification will bedescribed below.

(3) Forming Step

The pre-expanded beads are charged into a metal mold. Steam for heatingis introduced into the metal mold. The beads are foamed and fused by aforming machine into a shape of a product. In this step, the degree offoaming of the beads may be adjusted as required by a preliminarypressure device before forming.

(4) Drying Step

The resulting product is dried in a drying room at 40° C. to 60° C.

The experimental results of classification of the pre-expanded beadscontaining the filler will be described below.

1) Gravity Separation

Pre-expanded beads having the density distribution shown in FIG. 4 (airis used as the gas) were classified into eight types by a gravityseparator GA100 manufactured by Cimbria HEID GmbH under the conditionsas follows: vibration, 30 times/min; air 25 L/min: angle of inclination,A5.0°; and feed rate of beads: 9 kg/min. FIG. 5 shows densitydistribution of the beads in each classified lot.

2) Size Classification

Pre-expanded beads having a substantially constant filler content andhaving significantly low variations in weight were produced. Theresulting beads were classified through screens having different openingsizes (according to Japanese Industrial Standard Mesh 2.48, 2.38, 2.28,2.18, 2.08, 1.98, and 1.88). FIG. 6 shows the density distribution ofthe beads in each classified lot. In Comparative example 1 in the Table,the beads were not subjected to classification. In Comparative example12, workability was significantly poor since each specific gravity ofthe beads was measured by a methanol method.

The experimental results show that the classification and selection cansignificantly reduce the variations in the gas volume fraction that aregenerated during the foaming step, that are most difficult to control,and that cause nonuniformity of the dielectric constant.

Examples of lenses according to the present invention will now bedescribed.

A PP manufactured by Sumitomo Chemical Co., Ltd. and CaTiO₃ manufacturedby Otsuka Chemical Co., Ltd. were kneaded by a biaxial extruder. Afterkneading, the resulting resin mixture was substantially uniformly cutinto pellets each having a length of 2 mm with a pelletizer.

In the resulting pellets composed of the resin mixture, the weight ratioof polypropylene (PP) to CaTiO₃, i.e., PP/CaTiO₃, was 50/50, and thevariations were within 0.3 percent by weight.

Next, the resulting pellets were charged into a foaming tank. CO₂ wasenclosed in the pellets. Pre-expansion was then performed. The resultingpre-expanded beads were classified by size classification or gravityseparation. A preliminary pressure was applied to the preparedpre-expanded beads with a preliminary pressure device. The resultingbeads were charged into a metal mold. Steam was introduced into themetal mold to perform foaming.

Eight metal molds were used to produce eight types of foamed dielectricsegments (a pair of hemispherical bodies that were to be faced eachother and disposed at the center; and seven pairs of hemisphericalshells, each pair being a different type and shape from the other pair,which are to be laminated on the outer side of the inner side bodies).These were combined into a spherical Luneberg lens having a diameter of450 mm. For a layer having a dielectric constant of 1.5 or less,pre-expanded beads that were not subjected to classification was used.

The resulting spherical Luneberg lens 1, a receiving antenna A, and atransmitting antenna B were disposed in an anechoic chamber as shown inFIG. 3. The gain of the Luneberg lens 1, variations in gain, and anantenna pattern (sidelobe and the like) were measured. The table showsthe results. TABLE 1 Classification Produced Luneberg lens SeparationGain [dB] sidelobe Workability Method level Time Blending f(A) MaxVariations 32-25 29-25 (cost) 1 Comparative None 0.210 26.3 ±3.0 F F Pexample 2 Example Size 0.2 mm 1 None 0.020 32.5 ±1.0 P F P 3 ExampleSize 0.1 mm 1 None 0.015 33.1 ±0.8 P F P 4 Example Size 0.1 mm 2 None0.008 33.5 ±0.2 P P P 5 Example Size 0.1 mm 3 None 0.002 33.8 <±0.2 P PP 6 Example Size 0.1 mm 5 None 0.0008 34.0 <±0.2 P P G 7 Example Size0.05 mm  5 None 0.0006 34.0 <±0.2 P P G 8 Example Size 0.2 mm 1 Done0.08 32.0 ±1.3 P F P (adjoining peaks) 9 Example Size 0.1 mm 1 Done0.020 32.8 ±1.0 P F P (adjoining peaks) 10 Example Size 0.1 mm 2 Done0.015 33.3 ±0.5 P F P (adjoining peaks) 11 Comparative Size 0.2 mm 1Done 0.12 31.5 ±1.5 F F P example (nonadjoining peaks) 12 ComparativeDensity of each of beads 0.0004 34.0 <±0.2 P P F example was measured 13Example Gravity  6 1 None 0.019 32.5 ±1.0 P F P 14 Example Gravity  8 1None 0.015 33.1 ±0.8 P F P 15 Example Gravity 12 1 None 0.008 33.5 ±0.2P P P 16 Example Gravity 12 1 Done 0.014 33.2 ±0.6 P F P (adjoiningpeaks)Method: The term “size” represents size classification, and the term“gravity” represents gravity separation.Separation level: In size classification, the values represent the sizesof the openings of screens. In gravity separation, the values representthe# number of groups classified (the number of lots classified).Blending: The term “adjoining peaks” means that a bead group having adensity closest to and higher than a target density and a bead grouphaving a# density closest to and lower than the target density were mixed at apredetermined ratio to adjust # the density. The term “nonadjoiningpeaks” means that a bead group having a density second-closest to andhigher than a target density and a bead group # having a densitysecond-closest to and lower than the target density were mixed at apredetermined ratio to adjust the density.Sidelobe: The exceeding values of sidelobes are within 10% relative tothe specified values 32-25 logΦ in the first design example according toEIAJ# CPR-5104A or the specified values 29-25 logΦ according to ITU-Rrecommendation.Workability (cost): The process has industrial productivity.P: pass,G: good, andF: failure.

The table shows that when the classification and selection of thepre-expanded beads is performed, and thus f(A) represented by theequation f(A)=σa/Aave satisfies the expression 0.0005≦f(A)≦0.1, it ispossible to produce an electromagnetic wave lens having high gain,stability, and low sidelobes, and satisfying stringent requirements ofthe receiving antenna.

INDUSTRIAL APPLICABILITY

As has been described above, the content of a gas in at least adielectric foamed layer having a dielectric constant of 1.5 or more isuniformized to improve the uniformity of the dielectric constants.Therefore, it is possible to provide a Luneberg lens having high gainand low sidelobes. In particular, with respect to the sidelobes, it ispossible to provide the lens that has such small deviations as cansufficiently satisfy the stringent recommended values required for thereceiving antenna.

The G/T characteristic is also improved because of high gain and lowsidelobes.

Furthermore, high uniformity is achieved, and thus it is possible toensure characteristics essentially required for a multi-beam antenna,i.e., invariable gain, sidelobes, and focal length for electromagneticwaves from any direction.

The inorganic filler having a high dielectric constant is added, and theexpansion ratio is increased; hence, it is possible to provide alightweight lens.

In addition, according to the present invention, the forming canefficiently be performed: with a general-purpose bead-forming machine,enabling high mass productivity, good yield, and a reduction in the costof lenses.

1. A Luneberg lens having a single-layer structure or a multilayerstructure containing a plurality of layers having different dielectricconstants, wherein the respective structure is produced by mixing apolyolefin resin and/or a derivative thereof with an inorganic fillerhaving a high dielectric constant, the volume ratio of the polyolefinresin and/or the derivative thereof to the filler being 99 to 50:1 to50, adding a foaming agent to the resulting resin mixture and thenperforming preliminary expansion, and molding the resulting pre-expandedbeads; and wherein at least a foamed dielectric layer having adielectric constant of 1.5 or more is formed using the pre-expandedbeads that have been subjected to classification and selection such thatf(A) satisfies the expression 0.0005≦f(A)≦0.1, where f(A) is representedby the equation: f(A)=σa/Aave, σa is the deviation of a gas volumefraction Ar in the foamed dielectric layer, and Aave is the average ofthe gas volume fractions Ars at positions in the foamed dielectriclayer.
 2. The Luneberg lens according to claim 1, wherein the inorganicfiller having a high dielectric constant comprises titanium oxide, atitanate, a zirconate, or a mixture thereof.
 3. The Luneberg lensaccording to claim 2, wherein the titanate is barium titanate, strontiumtitanate, calcium titanate, or magnesium titanate.
 4. The Luneberg lensaccording to claim 1 or 2, wherein the foamed dielectric layer having adielectric constant of 1.5 or more is formed using the pre-expandedbeads classified by gravity separation or size classification.
 5. Amethod of producing a Luneberg lens that satisfies the requirementsdescribed in claim 1, comprising the steps of: mixing a polyolefin resinand/or a derivative thereof with an inorganic filler having a highdielectric constant, the volume ratio of the polyolefin resin and/or thederivative thereof to the filler being 99 to 50:1 to 50; adding afoaming agent to the resulting resin mixture and then performingpre-expansion; classifying and selecting the resulting pre-expandedbeads by gravity separation or size classification; and forming theclassified and selected pre-expanded beads into a shape.